As the second contribution, we show that the equivalence theory also covers function-valued functionals of the regression function, which hereafter are referred to as nonparametric functionals. As an important implication, we find GP priors enjoy a “nonparametric plug-in property” that generalizes the plug-in property coined by [1]} from real-valued to nonparametric functionals. A nonparametric estimator attains the plug-in property if it simultaneously achieves the minimax convergence rate for estimating the unknown function and the parametric \(1/\sqrt{n}\) -rate for estimating some bounded linear functionals. Although the plug-in property was originally proposed in the frequentist context, it holds for many Bayes estimators [2]}. However, (bounded) linear functionals considered in the plug-in property is restrictive in that the derivatives of a function at a fixed point are excluded from such functionals. In practice, the entire derivative function as opposed to its evaluation at a fixed point might be more of interest [3]}, [4]}, which to date is a largely unaddressed problem; one notable exception is [5]}, which studies a random series prior based on tensor product of B-splines. We extend the equivalence theory to the differential operators; in particular, this leads to a new approach to infer function derivatives in nonparametric regression. We show that the posterior distribution concentrates at such nonparametric functional at a nearly minimax rate [6]} in specific examples, thus achieving the nonparametric plug-in property. In addition, we find the rate-optimal estimation of \(f_0\)  and its derivatives share the same choice of hyperparameter in the GP prior, which is a remarkable property as it indicates the Bayes procedure automatically adapts to the order of derivative in this case. Finite sample performance of the proposed nonparametric plug-in procedure is assessed through simulations.
